Local coordinates for SL ( n , C ) -character varieties of finite-volume hyperbolic 3-manifolds

Pere Menal-Ferrer[1]; Joan Porti[1]

  • [1] Departament de Matemàtiques Universitat Autònoma de Barcelona, Bellaterra, Barcelona 08193, Spain

Annales mathématiques Blaise Pascal (2012)

  • Volume: 19, Issue: 1, page 107-122
  • ISSN: 1259-1734

Abstract

top
Given a finite-volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL ( 2 , C ) with the n -dimensional irreducible representation of SL ( 2 , C ) in SL ( n , C ) . In this paper we give local coordinates of the SL ( n , C ) -character variety around the character of this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.

How to cite

top

Menal-Ferrer, Pere, and Porti, Joan. "Local coordinates for $\operatorname{SL}(n,\mathbf{C})$-character varieties of finite-volume hyperbolic 3-manifolds." Annales mathématiques Blaise Pascal 19.1 (2012): 107-122. <http://eudml.org/doc/251117>.

@article{Menal2012,
abstract = {Given a finite-volume hyperbolic 3-manifold, we compose a lift of the holonomy in $\operatorname\{SL\}(2,\mathbf\{C\})$ with the $n$-dimensional irreducible representation of $\operatorname\{SL\}(2,\mathbf\{C\})$ in $\operatorname\{SL\}(n,\mathbf\{C\})$. In this paper we give local coordinates of the $\operatorname\{SL\}(n,\mathbf\{C\})$-character variety around the character of this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.},
affiliation = {Departament de Matemàtiques Universitat Autònoma de Barcelona, Bellaterra, Barcelona 08193, Spain; Departament de Matemàtiques Universitat Autònoma de Barcelona, Bellaterra, Barcelona 08193, Spain},
author = {Menal-Ferrer, Pere, Porti, Joan},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Infinitesimal Rigidity; Character Variety; Hyperbolic 3-Manifold; L2-Cohomology; hyperbolic 3–manifold; character variety; –cohomology; infinitesimal deformations},
language = {eng},
month = {1},
number = {1},
pages = {107-122},
publisher = {Annales mathématiques Blaise Pascal},
title = {Local coordinates for $\operatorname\{SL\}(n,\mathbf\{C\})$-character varieties of finite-volume hyperbolic 3-manifolds},
url = {http://eudml.org/doc/251117},
volume = {19},
year = {2012},
}

TY - JOUR
AU - Menal-Ferrer, Pere
AU - Porti, Joan
TI - Local coordinates for $\operatorname{SL}(n,\mathbf{C})$-character varieties of finite-volume hyperbolic 3-manifolds
JO - Annales mathématiques Blaise Pascal
DA - 2012/1//
PB - Annales mathématiques Blaise Pascal
VL - 19
IS - 1
SP - 107
EP - 122
AB - Given a finite-volume hyperbolic 3-manifold, we compose a lift of the holonomy in $\operatorname{SL}(2,\mathbf{C})$ with the $n$-dimensional irreducible representation of $\operatorname{SL}(2,\mathbf{C})$ in $\operatorname{SL}(n,\mathbf{C})$. In this paper we give local coordinates of the $\operatorname{SL}(n,\mathbf{C})$-character variety around the character of this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.
LA - eng
KW - Infinitesimal Rigidity; Character Variety; Hyperbolic 3-Manifold; L2-Cohomology; hyperbolic 3–manifold; character variety; –cohomology; infinitesimal deformations
UR - http://eudml.org/doc/251117
ER -

References

top
  1. Michael T. Anderson, Topics in conformally compact Einstein metrics, Perspectives in Riemannian geometry 40 (2006), 1-26, Amer. Math. Soc., Providence, RI Zbl1110.53031MR2237104
  2. K. Bromberg, Rigidity of geometrically finite hyperbolic cone-manifolds, Geom. Dedicata 105 (2004), 143-170 Zbl1057.53029MR2057249
  3. Michael Kapovich, Hyperbolic manifolds and discrete groups, 183 (2001), Birkhäuser Boston Inc., Boston, MA Zbl0958.57001MR1792613
  4. Alexander Lubotzky, Andy R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985) Zbl0598.14042MR818915
  5. Yozô Matsushima, Shingo Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365-416 Zbl0125.10702MR153028
  6. Pere Menal-Ferrer, Joan Porti, Twisted cohomology for hyperbolic three manifolds, to appear in Osaka J. Math. (2012), arXiv:1001.2242 Zbl1255.57018
  7. M. S. Raghunathan, On the first cohomology of discrete subgroups of semisimple Lie groups, Amer. J. Math. 87 (1965), 103-139 Zbl0132.02102MR173730
  8. André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149-157 Zbl0192.12802MR169956
  9. Hartmut Weiss, Local rigidity of 3-dimensional cone-manifolds, J. Differential Geom. 71 (2005), 437-506 Zbl1098.53038MR2198808

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.